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ISSN 1364-0380 (on line) 1465-3060 (printed)
Geometry & Topology
Volume 9 (2005) 2395–2415
Erratum 1
Published: 9 February 2005
Correction to: Construction of 2–local finite groups
of a type studied by Solomon and Benson
Ran Levi
Bob Oliver
Department of Mathematical Sciences, University of Aberdeen
Meston Building 339, Aberdeen AB24 3UE, UK
and
LAGA, Institut Galilée, Av. J-B Clément
93430 Villetaneuse, France
Email: ran@maths.abdn.ac.uk and
bob@math.univ-paris13.fr
Abstract A p–local finite group is an algebraic structure with a classifying
space which has many of the properties of p–completed classifying spaces
of finite groups. In our paper [2], we constructed a family of 2–local finite
groups which are “exotic” in the following sense: they are based on certain
fusion systems over the Sylow 2–subgroup of Spin7 (q) (q an odd prime
power) shown by Solomon not to occur as the 2–fusion in any actual finite
group. As predicted by Benson, the classifying spaces of these 2–local finite
groups are very closely related to the Dwyer–Wilkerson space BDI(4). An
error in our paper [2] was pointed out to us by Andy Chermak, and we
correct that error here.
Keywords Classifying space, p–completion, finite groups, fusion
AMS Classification 55R35; 55R37, 20D06, 20D20
A saturated fusion system over a finite p–group S is a category whose objects
are the subgroups of S , whose morphisms are all monomorphisms between the
subgroups, and which satisfy certain axioms first formulated by Puig, and also
described at the start of the first section in [2]. The main result of [2] is the
construction of saturated fusion systems over certain 2–groups, motivated by
a theorem of Solomon [3], which implies that these systems cannot be induced
by fusion in any finite group. Recently, Andy Chermak has pointed out to us
that the fusion systems actually constructed in [2] are not saturated (do not
satisfy all of Puig’s axioms). In this note, we describe how to modify that
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construction in a way so as to obtain saturated fusion systems of the desired
type, and explain why all of the results in [2] (aside from [2, Lemma A.10]) are
true under this new construction.
The following is the main theorem in [2]:
Theorem 1.1 [2, Theorem 2.1] Let q be an odd prime power, and fix S ∈
Syl2 (Spin7 (q)). Let z ∈ Z(Spin7 (q)) be the central element of order 2. Then
there is a saturated fusion system F = FSol (q) which satisfies the following
conditions:
(a) CF (z) = FS (Spin7 (q)) as fusion systems over S .
(b) All involutions of S are F –conjugate.
Furthermore, there is a unique centric linking system L = LcSol (q) associated
to F .
We have, in fact, found two errors in our proof of this theorem which we correct
here. The more serious one is in [2, Lemma A.10], which is not true as stated:
the last sentence in its proof is wrong. This has several implications on the
rest of our construction, all of which are systematically treated here. There is
also an error in the statement of [2, Lemma 2.8(b)] which is corrected below
(Lemma 1.8).
We first state and prove here a corrected version of [2, Lemma A.10], and
then state a modified version of the main technical proposition, [2, Proposition
1.2], used to prove saturation. Afterwards, we describe the changes which are
needed in [2, Section 2] to prove the main theorem. In table 1, we list the
correspondence between results and proofs in [2, Section 2] and those here.
This is intended as a guide to the reader who is not yet familiar with [2], and
who wants to read it simultaneously with this correction.
The only difference between [2, Lemma A.10] and the corrected version shown
here is that in [2], we claimed that the “correction factor” Z must lie in a certain
subgroup of order 2 in SL3 (Z/2k ), which is definitely not the case. Also, for
b2 ,
convenience, we state this lemma here for matrices over the 2–adic integers Z
k
instead of for matrices over the finite rings Z/2 .
Lemma 1.2 (Modified [2, Lemma A.10]) Let T1 and T2 be the two maximal
parabolic subgroups of GL3 (2):
T1 = GL12 (Z/2) = (aij ) ∈ GL3 (2) | a21 = a31 = 0
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Reference in [2]
Reference here
Remarks
Proposition 1.2
Theorem 2.1
Definition 2.2, Lemma 2.3
Definition 2.4
Prp. 2.5, Def. 2.6, Lem. 2.7
—
—
—
Lemma 2.8
Proposition 1.3
Theorem 1.1
—
—
—
Lem. 1.4, Prp. 1.5
Definition 1.6
Lemma 1.7
Lemma 1.8
Proposition 2.9
Lemma 2.10
Proposition 2.11
Proposition 1.9
Lemma 1.10
Proposition 1.11
more general statement
unchanged
unchanged
omit definition of Γn
unchanged
added
new definition of Γn
added
(b) restated,
new proofs of (b), (e)
partly new proof
partly new proof
partly new proof
Table 1
and
T2 = GL21 (Z/2) = (aij ) ∈ GL3 (2) | a31 = a32 = 0 .
Set T0 = T1 ∩ T2 : the group of upper triangular matrices in GL3 (2). Assume,
for some k ≥ 2, that
b 2)
µi : Ti −−−−−→ SL3 (Z
are lifts of the inclusions Ti −−→ GL3 (2) = SL3 (2) (for i = 1, 2) such that
µ1 |T0 = µ2 |T0 . Then there is a homomorphism
b 2 ),
µ : GL3 (2) −−−→ SL3 (Z
and an element Z ∈ CSL (bZ ) (µ1 (T0 )), such that µ|T1 = µ1 , and µ|T2 = cZ ◦ µ2 .
3 2
b 2 ) of the
→ SL3 (Z
Proof By [1, Lemma 4.4], there is a lifting µ′ : GL3 (2)
b
identity on GL3 (2); and any two liftings to SL3 (Z2 ) of the inclusion of T1 or of
b 2 ). In particular,
T2 into GL3 (2) differ by conjugation by an element of SL3 (Z
b
there are elements Z1 , Z2 ∈ SL3 (Z2 ) such that µi = cZi ◦ µ′ |Ti (for i = 1, 2).
Set µ = cZ1 ◦ µ′ , and Z = Z1 Z2−1 . Then µ1 = µ|Ti , and µ|T2 = cZ ◦ µ2 . Since
µ1 |T0 = µ2 |T0 , conjugation by Z is the identity on µ1 (T0 ) = µ2 (T0 ), and thus
Z ∈ CSL (bZ ) (µ1 (T0 )).
3
2
Whenever G is a finite group, S ∈ Sylp (G), S0 ⊳ S , and Γ ≤ Aut(S0 ), then
hFS (G); FS0 (Γ)i
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denotes the smallest fusion system over S which contains all G–fusion, and
which also contains all restrictions of automorphisms in Γ. In other words, if
F denotes this fusion system, then for each P, Q ≤ S , HomF (P, Q) is the set
of all composites
ϕ1
ϕ2
ϕ
P = P0 −−−→ P1 −−−→ P2 −−−→ · · · −−−→ Pk−1 −−−k→ Pk = Q,
where for each i, Pi ≤ S , and either ϕi ∈ HomG (Pi−1 , Pi ), or (if Pi−1 , Pi ≤ S0 )
ϕi = ψi |Pi−1 for some ψi ∈ Γ such that ψi (Pi−1 ) = Pi .
Whenever G is a finite group and S ∈ Sylp (G), an automorphism ϕ ∈ Aut(S)
is said to preserve G–fusion if for each P, Q ≤ S and each α ∈ Iso(P, Q),
α ∈ IsoG (P, Q) if and only if ϕαϕ−1 ∈ IsoG (ϕ(P ), ϕ(Q)). The proof of Theorem
1.1 is based on the following proposition.
Proposition 1.3 (Modified [2, Proposition 1.2]) Fix a finite group G, a prime
p dividing |G|, and a Sylow p–subgroup S ∈ Sylp (G). Fix a normal subgroup
Z ⊳ G of order p, an elementary abelian subgroup U ⊳ S of rank two containing Z such that CS (U ) ∈ Sylp (CG (U )), and a group Γ ≤ Aut(CS (U )) of
automorphisms which preserve all CG (U )–fusion, and such that γ(U ) = U for
all γ ∈ Γ. Set
S0 = CS (U )
and
F = hFS (G); FS0 (Γ)i,
and assume the following hold.
(a) All subgroups of order p in S different from Z are G–conjugate.
(b) Γ permutes transitively the subgroups of order p in U .
(c) {ϕ ∈ Γ | ϕ(Z) = Z} = AutNG (U ) (CS (U )).
(d) For each E ≤ S which is elementary abelian of rank three, contains U ,
and is fully centralized in FS (G),
{α ∈ AutF (CS (E)) | α(Z) = Z} = AutG (CS (E)).
(e) For all E, E ′ ≤ S which are elementary abelian of rank three and contain
U , if E and E ′ are Γ–conjugate, then they are G–conjugate.
Then F is a saturated fusion system over S . Also, for any P ≤ S such that
Z ≤ P,
{ϕ ∈ HomF (P, S) | ϕ(Z) = Z} = HomG (P, S).
(1)
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This proposition is slightly more general than [2, Proposition 1.2], in that Γ
is assumed only to be a group of automorphisms of CS (U ) which preserves
CG (U )–fusion, and not a group of automorphisms of CG (U ) itself. This extra
generality is necessary when proving that the fusion systems FSol (q), under our
modified construction, are saturated. The changes needed to prove this more
general version of [2, Proposition 1.2] are described in Section 2.
Proposition 1.3 is applied with G = Spin7 (q), Z = Z(G), and S ∈ Syl2 (G),
U ≤ S , and Γ ≤ Aut(CG (U )) to be chosen shortly. The error in [2] arose in
the choice of Γ, as will be explained in detail below.
We first recall some of the definitions and notation used in [2]. Throughout,
we fix an odd prime power q , let Fq be a field with q elements, and let Fq be
its algebraic closure. We write SL2 (q ∞ ) = SL2 (Fq ), Spin7 (q ∞ ) = Spin7 (Fq ),
n
etc., for short. For each n, ψ q denotes the automorphism of Spin7 (q ∞ ) or of
n
SL2 (q ∞ ) induced by the field isomorphism (x 7→ xq ). We then fix elements
z, z1 ∈ Spin7 (q) of order 2, where hzi = Z(Spin7 (q)), set U = hz, z1 i, and
construct an explicit homomorphism
ω : SL2 (q ∞ )3 −−−−−→ Spin7 (q ∞ )
such that
Im(ω) = CSpin7 (q∞ ) (U )
and
Ker(ω) = h(−I, −I, −I)i.
Write
def
H(q ∞ ) = ω(SL2 (q ∞ )3 ) = CSpin7 (q∞ ) (U )
and [[X1 , X2 , X3 ]] = ω(X1 , X2 , X3 )
for short. In particular,
z = [[I, I, −I]]
and thus
and
z1 = [[−I, I, I]],
U = [[±I, ±I, ±I]]
(with all combinations of signs). By [2, Lemma 2.3 & Proposition 2.5], there is
an element τ ∈ NSpin7 (q) (U ) of order 2 such that
τ ·[[X1 , X2 , X3 ]]·τ −1 = [[X2 , X1 , X3 ]]
for all X1 , X2 , X3 ∈ SL2 (q ∞ ), and such that
NSpin7 (q∞ ) (U ) = H(q ∞ )·hτ i.
We next fix elements A, B ∈ SL2 (q) of order 4, such that hA, Bi ∼
= Q8 (a
quaternion group of order 8). Most of the following notation is taken from [2,
Definition 2.6]. We set
Ab = [[A, A, A]]
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and
b = [[B, B, B]];
B
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k
C(q ∞ ) = {X ∈ CSL2 (q∞ ) (A) | X 2 = I, some k } ∼
= Z/2∞ ;
and
Q(q ∞ ) = hC(q ∞ ), Bi.
Here, Z/2∞ means a union of cyclic 2–groups Z/2n for all n; equivalently, the
group Z[ 12 ]/Z. We then define
A(q ∞ ) = ω(C(q ∞ )3 ) ∼
= (Z/2∞ )3 ,
S0 (q ∞ ) = ω(Q(q ∞ )3 ) ≤ H(q ∞ )
S(q ∞ ) = S0 (q ∞ )·hτ i ≤ H(q ∞ )·hτ i ≤ Spin7 (q ∞ ).
In all cases, whenever a subgroup Θ(q ∞ ) ≤ Spin7 (q ∞ ) has been defined, we set
Θ(q n ) = Θ(q ∞ ) ∩ Spin7 (q n ).
n
Since Spin7 (q n ) is the fixed subgroup of ψ q acting on Spin7 (q ∞ ) (cf. [2,
Lemma A.3]), H(q n ) ≤ H(q ∞ ) is the subgroup of all elements of the form
n
[[X1 , X2 , X3 ]], where either Xi ∈ SL2 (q n ) for each i, or ψ q (Xi ) = −Xi for
each i. By [2, Lemma 2.7], for all n,
S(q n ) ∈ Syl2 (Spin7 (q n )).
The following lemma is what is needed to tell us how to choose a subgroup
Γn ≤ Aut(S0 (q n )) so that the fusion system hFS(qn ) (Spin7 (q n )); FS0 (qn ) (Γn )i is
saturated. Note that since each element of C(q ∞ ) has 2–power order, it makes
b2 .
sense to write X u ∈ C(q ∞ ) for X ∈ C(q ∞ ) and u ∈ Z
Lemma 1.4 Assume α ∈ Aut(A(q ∞ )) centralizes AutS(q∞ ) (A(q ∞ )). Then α
has the form
α([[X1 , X2 , X3 ]]) = [[X1v , X2v , X3u ]]
b 2 )∗ .
for some u, v ∈ (Z
Proof Set
∆0 = AutS(q∞ ) (A(q ∞ )) = hc[[B,I,I]], c[[I,B,I]], c[[I,I,B]], cτ i
for short. The second equality follows since S(q ∞ ) is by definition generated
by A(q ∞ ) and the four elements listed. Set
def
b ∼
A1 = hz, z1 , Ai
= C23 ,
the 2–torsion subgroup in A(q ∞ ). The image of ∆0 ≤ Aut(A(q ∞ )) in the
group Aut(A1 ) ∼
= GL3 (2) (the image under restriction) is the group of all
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automorphisms which leave hzi and U = hz, z1 i invariant (ie, the group of
b
upper triangular matrices with respect to the ordered basis {z, z1 , A}).
By assumption, [α, ∆0 ] = 1, and in particular, α∆0 α−1 = ∆0 . Since each
element of ∆0 sends U to itself, this means that each element of ∆0 also sends
α(U ) to itself. Also, U is the only subgroup of rank 2 left invariant by all
elements of ∆0 (since ∆0 contains all automorphisms which leave hzi and U
invariant), and hence α(U ) = U .
It follows that α induces an automorphism α′ of A(q ∞ )/U = (C(q ∞ )/h − Ii)3 .
Also, α′ commutes with the following automorphisms of (C(q ∞ )/h − Ii)3 :
(Y1 , Y2 , Y3 ) 7→ (Y1±1 , Y2±1 , Y3±1 )
and
(Y1 , Y2 , Y3 ) 7→ (Y2 , Y1 , Y3 )
since these are induced by automorphisms in ∆0 . Hence α′ has the form
b 2 )∗ . Thus for all X1 , X2 , X3 ∈
α′ (Y1 , Y2 , Y3 ) = (Y1v , Y2v , Y3u ) for some u, v ∈ (Z
C(q ∞ ),
α([[X1 , X2 , X3 ]]) = [[±X1v , ±X2v , ±X3u ]].
Since all elements of C(q ∞ ) are squares, these signs must all be positive, and
α has the form α([[X1 , X2 , X3 ]] = [[X1v , X2v , X3u ]].
b 2 )∗ , let δu ∈ Aut(A(q ∞ )) be the automorphism
For each u ∈ (Z
δu ([[X1 , X2 , X3 ]]) = [X1 , X2 , X3u ].
Define γ, γu ∈ Aut(A(q ∞ )) by setting
γ([[X1 , X2 , X3 ]]) = [[X3 , X1 , X2 ]]
and
γu = δu γδu−1 .
b 2 such that u ≡ 1 (mod 4), and
Proposition 1.5 There is an element u ∈ Z
∞
such that the subgroup Ωu ≤ Aut(A(q )) given by
def
Ωu = h AutSpin7 (q∞ ) (A(q ∞ )), γu i
is isomorphic to C2 × GL3 (2). Furthermore, the following hold:
(a) The subgroup of elements of Ωu which act via the identity on all 2–torsion
in A(q ∞ ) has order 2, and contains only the identity and the automorphism (g 7→ g−1 ).
(b) For each n ≥ 1,
h AutSpin7 (qn ) (A(q n )), γu i ∼
= C2 × GL3 (2).
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Proof For each k ≥ 1, let Ak ≤ A(q ∞ ) denote the 2k –torsion subgroup. In
particular,
b ∼
A1 = hz, z1 , Ai
= C23 ,
where
Ab = [[A, A, A]].
Let R1 , R2 , · · · ∈ C(q ∞ ) ∼
= Z/2∞ be elements such that R1 = −I , R2 = A,
2
and (Ri ) = Ri−1 for all k ≥ 2. Thus, |Ri | = 2i for all i. For each k ≥ 1, let
(k) (k) (k)
{r1 , r2 , r3 } be the basis of Ak defined by
(k)
r1 = [[I, I, Rk ]],
(k)
r2 = [[Rk , I, I]],
(1)
(1)
and
(k)
r3 = [[Rk+1 , Rk+1 , Rk+1 ]].
(1)
b Using these bases, we identify
In particular, r1 = z , r2 = z1 , and r3 = A.
k
∞
b 2 ).
Aut(Ak ) = GL3 (Z/2 ) and Aut(A(q )) = GL3 (Z
Set
∆0 = AutS(q∞ ) (A(q ∞ )),
∆1 = AutSpin7 (q∞ ) (A(q ∞ )),
and
∆2 = h∆0 , γi.
In particular, ∆2 is the group of all signed permutations
±1
±1
±1
[[X1 , X2 , X3 ]] 7→ [[Xσ(1)
, Xσ(2)
, Xσ(3)
]]
for σ ∈ Σ3 .
(k)
For each i = 0, 1, 2 and each k ≥ 1, let ∆i ≤ Aut(Ak ) be the image of ∆i
(1)
under restriction. By [2, Proposition A.8], ∆1 = AutSpin (A1 ) is the group of
(1)
all elements of Aut(A1 ) ∼
= GL3 (Z/2) which send z to itself. Also, ∆0 was seen
in the proof of Lemma 1.4 to be the group of all automorphisms of A1 which
leaves both z and U = hz, z1 i invariant; and a similar argument shows that
(1)
∆2 is the group of all automorphisms of A1 which leaves U invariant. Hence,
b of A1 , each group ∆(1) ≤ Aut(A1 )
with respect to the ordered basis {z, z1 , A}
i
(i = 0, 1, 2) can be identified with the subgroup Ti ≤ GL3 (Z/2) of Lemma 1.2.
By [2, Proposition 2.5],
CSpin7 (q∞ ) (U ) = H(q ∞ ) ∼
= (SL2 (q ∞ )3 )/h(−I, −I, −I)i.
def
An element [[X1 , X2 , X3 ]] ∈ H(q ∞ ) (Xi ∈ SL2 (q ∞ )) centralizes Ab = [[A, A, A]]
if and only if [Xi , A] = 1 for each i, or Xi AXi−1 = −A = A−1 for each i.
Set C = CSL2 (q∞ ) (A). This is an abelian group (the union of the finite cyclic
b
groups CSL2 (qn ) (A)), and NSL2 (q∞ ) (A) = C·hBi. Hence, since A1 = hU, Ai
b
and B = [[B, B, B]], we have
b = ω(C 3 )·hBi.
b
CSpin7 (q∞ ) (A1 ) = CH(q∞ ) (A)
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Since ω(C 3 ) is abelian (thus centralizes A(q ∞ )), this shows that the kernel of
(1)
b;
each of the projection maps ∆i −−։ ∆i is generated by conjugation by B
−1
ie, by the automorphism (g 7→ g ).
Since each ∆i is finite, their elements all have determinant of finite order in
b 2 )∗ , hence are ±1 in all cases. Also, for each i, ∆i surjects onto ∆(1) = Ti
(Z
i
with kernel generated by the automorphism (g 7→ g−1 ) of determinant (−1).
Hence the elements of determinant one in ∆i are sent isomorphically to Ti , and
define a lift
b 2)
µi : Ti −−−−−−→ SL3 (Z
with respect to the given bases. In particular, µ1 |T0 = µ2 |T0 = µ0 .
∞
b 2 )∗ , we define ψ
For all i, j, k ∈ (Z
i,j,k ∈ Aut(A(q )) by setting
ψi,j,k ([[X1 , X2 , X3 ]]) = [[X1i , X2j , X3k ]]
for all [[X1 , X2 , X3 ]] ∈ A(q ∞ ). Recall that cτ ([[X1 , X2 , X3 ]]) = [[X2 , X1 , X3 ]] (by
choice of τ ). Since ∆0 = AutS(q∞ ) (A(q ∞ )) is generated by c[[B,I,I]] , c[[I,B,I]] ,
c[[I,I,B]] , and cτ (corresponding to generators of S(q ∞ )/A(q ∞ )), µ0 has image
Im(µ0 ) = µ1 (T0 ) = hψ−1,−1,1 , ψ1,−1,−1 , cτ i,
where ψ−1,−1,1 = c[[B,B,I]] and ψ1,−1,−1 = c[[I,B,B]] .
By Lemma 1.2, there is a homomorphism
b 2 ),
µ : GL3 (2) −−−−−−→ SL3 (Z
b 2 ) which commutes with all elements
and an element Z ∈ Aut(A(q ∞ )) ∼
= GL3 (Z
of µ1 (T0 ), such that µ|T1 = µ1 and µ|T2 = cZ ◦ µ2 . By Lemma 1.4, Z = ψv,v,u
b 2 )∗ . Since ψv,v,v lies in the center
(using the above notation) for some u, v ∈ (Z
∞
of Aut(A(q )) (it sends every element to its v -th power), we can assume that
v = 1 (without changing cZ ), and thus that Z = ψ1,1,u = δu . Finally, since
δ−1 = ψ1,1,−1 ∈ ∆0 , we can replace δu by δ−u if necessary, and assume that
u ≡ 1 (mod 4).
b 2 ) = Aut(A(q ∞ )), we now have
Under the identification GL3 (Z
AutSpin7 (q∞ ) (A(q ∞ )) = (g 7→ g−1 ) × µ(T1 )
and
γu = δu γδu−1 = cZ (γ) = µ
0 1 0 110
001
,
(1)
(1)
(1)
where the matrix is that of γ|A1 with respect to the basis {r1 , r2 , r3 }. Also,
T1 is a maximal subgroup of GL3 (2) — the subgroup of invertible matrices
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(1)
which send r1 to itself — and so T1 together with this matrix generate GL3 (2).
Thus
Ωu = h AutSpin7 (q∞ ) (A(q ∞ )), γu i = hg 7→ g−1 i× µ(GL3 (2)) ∼
= C2 × GL3 (2). (2)
This proves the first claim in the proposition. Point (a) follows by construction,
since each nonidentity element of µ(GL3 (2)) acts nontrivially on A1 . Point (b)
follows from (2), once we know that each element of Spin7 (q n ) which normalizes
A1 (hence which normalizes A(q n )) also normalizes A(q ∞ ) — and this follows
from (1).
We are now in a position to define the fusion systems we want. Roughly, they
are generated by the fusion systems of Spin7 (q n ) together with one extra automorphism: the cyclic permutation [[X1 , X2 , X3 ]] 7→ [[X3 , X1 , X2 ]] “twisted” by
the automorphism δu of the last proposition. By comparison, the construction
in [2] was similar but without the twisting (ie, done with u = 1), and the
resulting fusion system is, in fact, not saturated.
We regard Q(q ∞ ) = C(q ∞ )⋊hBi as an infinite quaternion group: BA′ B −1 =
A′−1 for each A′ ∈ C(q ∞ ), and each element of the coset C(q ∞ )·B has order
4. Hence any automorphism of C(q ∞ ) extends to a unique automorphism of
Q(q ∞ ) which sends B to itself.
b 2 )∗ be as in Proposition 1.5. Let
Definition 1.6 Let u ∈ (Z
be the automorphisms
and
γb , δbu ∈ Aut(S0 (q ∞ ))
γb ([[X1 , X2 , X3 ]]) = [[X3 , X1 , X2 ]]
δbu ([[X1 , X2 , A′ B j ]]) = [[X1 , X2 , (A′ )u B j ]]
for all Xi ∈ Q(q ∞ ), A′ ∈ C(q ∞ ), and i, j ∈ Z; and set γbu = δbu γb δbu−1 ∈
Aut(S0 (q ∞ )). For each n ≥ 1, set
and set
Γn = h Inn(S0 (q n )), cτ , γbu i ≤ Aut(S0 (q n ));
Fn = FSol (q n ) = hFS(qn ) (Spin7 (q n )), FS0 (qn ) (Γn )i.
In order to be able to apply Proposition 1.3, it is important to know that Γn
is fusion preserving. This follows immediately from the following lemma.
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Lemma 1.7 For each n ≥ 1, the automorphisms cτ , γb , δbu , and γbu all preserve
H(q n )–fusion in S0 (q n ).
Proof The fusion in SL2 (q n ) is generated by inner automorphisms of its Sylow
subgroup Q(q n ), together with the groups Aut(P ) for subgroups P ≤ Q(q n )
isomorphic to Q8 . This follows from Alperin’s fusion theorem, since these are
the only subgroups whose automorphism group is not a 2–group. Thus any
automorphism of Q(q n ) — in particular, the automorphism ϕu defined by
ϕu (A′ ) = A′u , ϕu (B) = B for A′ ∈ C(q n ) — preserves fusion.
Set H0 (q n ) = {[[X1 , X2 , X3 ]] | Xi ∈ SL2 (q n ), ∀i}. Fix a generator Y of C(q 2n );
n
then C(q n ) = hY 2 i, and ψ q (Y ) = −Y . For each g = [[X1 , X2 , X3 ]] ∈ H(q n ),
n
n
n
[[ψ q (X1 ), ψ q (X2 ), ψ q (X3 )]] = [[X1 , X2 , X3 ]]; and hence there is some fixed
n
ǫ = ±1 for which ψ q (Xi ) = ǫ·Xi for i = 1, 2, 3. When ǫ = 1, this means
that Xi ∈ SL2 (q n ) for each i; while if ǫ = −1 it means that Xi ∈ SL2 (q n )·Y
for each i. So every element of H(q n ) either lies in H0 (q n ), or has the form
g·[[Y, Y, Y ]] for some g ∈ H0 (q n ).
Assume g ∈ H(q n ) and P, Q ≤ S0 (q n ) are such that gP g−1 = Q. Clearly,
P ≤ H0 (q n ) if and only if Q ≤ H0 (q n ) (H0 (q n ) is normal in H(q n )). We claim
that there is h ∈ H(q n ) such that
hδbu (P )h−1 = δbu (Q)
and
ch ◦ δbu |P = δbu ◦ cg |P .
(1)
Let Pi , Qi ≤ SL2 (q 2n ) be the projections of P and Q to the i-th factor (i =
1, 2, 3), and write g = [[X1 , X2 , X3 ]] (thus Xi Pi Xi−1 = Qi ). Consider the
following cases.
(a) Assume g ∈ H0 (q n ) and P, Q ≤ H0 (q n ). Since ϕu preserves fusion in
SL2 (q n ), there is Y3 ∈ SL2 (q n ) such that Y3 ϕu (P3 )Y3−1 = ϕu (Q3 ) and
def
cY3 ◦ ϕu |P3 = ϕu ◦ cX3 |P3 . Then h = [[X1 , X2 , Y3 ]] satisfies (1).
(b) Assume g ∈
/ H0 (q n ) and P, Q ≤ H0 (q n ). Write g = g′ ·[[Y, Y, Y ]], where
g′ ∈ H0 (q n ). Choose h′ as in (a), so that (1) is satisfied with g, h replaced
by g′ , h′ . Then the element h = h′ ·[[Y, Y, Y u ]] satisfies (1).
(c) Finally, assume that P, Q H0 (q n ). Then none of the subgroups Pi , Qi ≤
SL2 (q 2n ) is contained in SL2 (q n ). By the same procedure as was used in
(a), we can find h ∈ H0 (q 2n ) which satisfies (1); the problem is to do this
so that h ∈ H(q n ).
As noted above, fusion in SL2 (q 2n ) is generated by inner automorphisms
of Q(q 2n ) and automorphisms of subgroups isomorphic to Q8 . Hence if Pi
Geometry & Topology, Volume 9 (2005)
2406
is not isomorphic to Q8 or one of its subgroups, then there is Xi′ ∈ Q(q 2n )
such that cXi |Pi = cXi′ |Pi . If, on the other hand, Pi (and hence Qi ) is
isomorphic to Q8 or C4 , then Pi ≤ hA, Y r Bi and Qi ≤ hA, Y s Bi for
def
some odd r, s ∈ Z, and we can choose k ∈ Z such that Xi′ = Y k ∈ Q(q 2n )
has the same conjugation action as Xi . Thus in all cases, we can write
Xi = Xi′ Xi′′ for some Xi′ ∈ Q(q 2n ) and some Xi′′ ∈ CSL2 (q2n )·hY i (Pi ).
The subgroups of Q(q 2n ) which are centralized by elements in the coset
SL2 (q n )·Y are precisely the cyclic subgroups. (The quaternion subgroups
of order ≥ 8 are all centric in SL2 (q 2n ).) Hence we can choose elements
Yi′′ as follows: Yi′′ = 1 if Xi′′ ∈ SL2 (q n ), and Yi′′ ∈ SL2 (q n )·Y and
centralizes ϕu (Pi ) if Xi′′ ∈ SL2 (q n )·Y . We now define
h = [[X1′ , X2′ , ϕu (X3′ )]]·[[Y1′′ , Y2′′ , Y3′′ ]];
then h ∈ H(q n ) and satisfies (1).
This shows that δbu preserves H(q n )–fusion as an automorphism of S0 (q n ).
Also, γb and cτ preserve fusion, since both extend to automorphisms of H(q n );
and hence γbu = δbu γb δbu−1 also preserves fusion.
n
Let ψ = ψ q ∈ Aut(Spin7 (q ∞ )) be induced by the field automorphism x 7→
n
xq . By [2, Proposition A.9(a)], if E ≤ Spin7 (q n ) is an arbitrary elementary
abelian 2–subgroup of rank 4, then there is an element a ∈ Spin7 (q ∞ ) such that
aEa−1 = E∗ , and we define
xC (E) = a−1 ψ(a).
Then xC (E) ∈ E , and is independent of the choice of a.
In the following lemma, we correct the statement and proof of points (b) and
(e). The proof of (e) is affected by both the changes in the statement of (b)
and those in the definition of Γn .
b Bi
b ≤ S(q n ),
Lemma 1.8 [2, Lemma 2.8] Fix n ≥ 1, set E∗ = hz, z1 , A,
and let C be the Spin7 (q n )–conjugacy class of E∗ . Let E4U be the set of all
elementary abelian subgroups E ≤ S(q n ) of rank 4 which contain U = hz, z1 i.
Fix a generator X ∈ C(q n ) (the 2–power torsion in CSL2 (qn ) (A)), and choose
Y ∈ C(q 2n ) such that Y 2 = X . Then the following hold.
(a) E∗ has type I.
Geometry & Topology, Volume 9 (2005)
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(b) Each subgroup in E4U which contains Ab is of the form
b [[X i B, X j B, X k B]]i or
Eijk = hz, z1 , A,
′
b [[X i Y B, X j Y B, X k Y B]]i.
Eijk
= hz, z1 , A,
Each subgroup in E4U is H(q n )–conjugate to one of these subgroups Eijk
′
or Eijk
for some i, j, k ∈ Z.
′ ) = [[(−I)i , (−I)j , (−I)k ]]·A.
b
(c) xC (Eijk ) = [[(−I)i , (−I)j , (−I)k ]] and xC (Eijk
′
(d) All of the subgroups Eijk
have type II. The subgroup Eijk has type I if
and only if i ≡ j (mod 2), and lies in C (is conjugate to E∗ ) if and only if
i ≡ j ≡ k (mod 2). The subgroups E000 , E001 , and E100 thus represent
the three conjugacy classes of rank four elementary abelian subgroups of
Spin7 (q n ) (and E∗ = E000 ).
(e) For any ϕ ∈ Γn ≤ Aut(S0 (q n )) and any E ∈ E4U , ϕ(xC (E)) = xC (ϕ(E)).
Proof We prove only points (b) and (e) here, and refer to [2] for the proofs of
the other points.
b
(b) Assume first that Ab ∈ E ; ie, that E ≥ A1 = hz, z1 , Ai.
By definition,
n
n
S(q ) is generated by A(q ), whose elements clearly centralize A1 ; and elements
[[B i , B j , B k ]] for i, j, k ∈ {0, 1}. Since an element of this form centralizes Ab only
if i = j = k , this shows that
b = A(q n )·hBi.
b
E ≤ CS(qn ) (hz, z1 , Ai)
b g Bi
b
Since A(q n ) is a finite abelian 2–group of rank 3, we have E = hz, z1 , A,
n
for some g ∈ A(q ). Also,
A(q n ) = ω(C(q ∞ )3 ) ∩ Spin7 (q n ) = [[X i , X j , X k ]], [[X i Y, X j Y, X k Y ]] i, j, k ∈ Z ,
′
and hence E = Eijk or Eijk
for some i, j, k ∈ Z.
Now let E ∈ E4U be arbitrary. Each element of E has the form [[X1 , X2 , X3 ]],
where either Xi ∈ SL2 (q n ) for all i, or Xi ∈ SL2 (q n )·Y for all i — and the
elements of the first type (Xi ∈ SL2 (q n )) form a subgroup of index at most 2.
Since U has index 4 in E , this means that there is some g = [[X1 , X2 , X3 ]] ∈
ErU for which Xi ∈ SL2 (q n ) for all i. Also, |Xi | = 4 for all i, since g ∈
/
U = {[[±I, ±I, ±I]]}. Since all elements of order 4 in SL2 (q n ) are conjugate
b and hence that
(cf. [4, 3.6.23]), this implies that g is H(q n )–conjugate to A,
n
′ .
E is H(q )–conjugate to one of the above subgroups Eijk or Eijk
Geometry & Topology, Volume 9 (2005)
2408
(e) By construction, xC (−) is preserved under conjugation by elements of the
group Spin7 (q n ). Since Γn is generated by γbu and conjugation by elements of
Spin7 (q n ), it suffices to prove the result when ϕ = γbu . Since u ≡ 1 (mod 4) by
′ ) = E′
b = A,
b γ
bu (Eijk
bu (Eijk ) = Ek′ ,i′ ,j ′ , and γ
Proposition 1.5, γbu (A)
k ′′ ,i′′ ,j ′′ for
′
′′
some i ≡ i ≡ i (mod 2), and similarly for the other indices. Hence by (c),
′
γbu (xC (E)) = xC (γbu (E)) whenever E = Eijk or E = Eijk
for some i, j, k ∈ Z.
′ . By (b), there
Now assume E ∈ E4U is not one of the subgroups Eijk or Eijk
def
is g ∈ H(q n ) such that E ′ = gEg−1 is of this form. Since γbu preserves
H(q n )–fusion by Lemma 1.7, there is h ∈ H(q n ) such that the following square
commutes:
E
b
γu
→ γbu (E)
c
cg
h
↓
↓
b
γ
u
′
→ γbu (E ′ ) .
E
We have seen that γbu (xC (E ′ )) = xC (γbu (E ′ )); and also that cg and ch preserve
xC (−). Hence γbu (xC (E)) = xC (γbu (E)) by the commutativity of the square.
The following is the crucial result needed to apply Proposition 1.3. The statement is exactly the same as that in [2], but the proof has to be modified slightly
due to the changed definition of Γn (hence of Fn ).
Proposition 1.9 [2, Proposition 2.9] Fix n ≥ 1. Let E ≤ S(q n ) be an elementary abelian subgroup of rank 3 which contains U , and such that CS(qn ) (E)
∈ Syl2 (CSpin7 (qn ) (E)). Then
{ϕ ∈ AutFn (CS(qn ) (E)) | ϕ(z) = z} = AutSpin7 (qn ) (CS(qn ) (E)).
(1)
Proof Set
Spin = Spin7 (q n ),
S = S(q n ),
Γ = Γn ,
and
F = Fn
for short, and consider the subgroups
def
R0 = R0 (q n ) = A(q n )
Then
and
R0 ∼
= (C2k )3
def
b = hR0 , Bi.
b
R1 = R1 (q n ) = CS (hU, Ai)
and
b
R1 = R0 ⋊ hBi,
b = [[B, B, B]]
where 2k is the largest power which divides q n ± 1, and where B
−1
has order 2 and acts on R0 via (g 7→ g ). Also,
b = h[[±I, ±I, ±I]], [[A, A, A]]i ∼
hU, Ai
= C3
2
Geometry & Topology, Volume 9 (2005)
2409
is the 2–torsion subgroup of R0 . It was shown in the proof of [2, Proposition
2.9] that
(2)
R0 is the only subgroup of S isomorphic to (C2k )3 .
Let E ≤ S be an elementary abelian subgroup of rank 3 which contains U ,
and such that CS (E) ∈ Syl2 (CSpin (E)). There are two cases to consider: that
where E ≤ R0 and that where E R0 .
Case 1 Assume E ≤ R0 . Since R0 is abelian of rank 3, we must have
b the 2–torsion subgroup of R0 , and CS (E) = R1 . Also, by (2),
E = hU, Ai,
neither R0 nor R1 is isomorphic to any other subgroup of S ; and hence
for i = 0, 1.
(3)
(Recall that AutΓ (Ri ) is generated by AutS0 (qn ) (Ri ) and the restrictions of cτ
and γu .) Hence by Proposition 1.5(b),
AutF (Ri ) = AutSpin (Ri ), AutΓ (Ri ) = AutSpin (Ri ), γu |Ri
AutF (R0 ) = AutSpin (R0 ), γu |R0 ∼
= C2 × GL3 (2).
b denote the 2–torsion subgroup of R0 ,
In other words, if we let A1 = hz, z1 , Ai
then restriction to A1 sends AutF (R0 ) onto Aut(A1 ) with kernel hcBb i of order
2. Since AutSpin (A1 ) is the group of all automorphisms of A1 which send z to
itself [2, Proposition A.8], this shows that
AutSpin (R0 ) = ϕ ∈ AutF (R0 ) ϕ(z) = z .
(4)
In the proof of [2, Proposition 2.9] (see formula (5) in that proof), we show the
first of the following two equalities:
b =B
b
AutSpin (R1 ) = Inn(R1 )· ϕ ∈ AutSpin (R1 ) ϕ(B)
b = B,
b ϕ|R ∈ AutSpin (R0 ) . (5)
= Inn(R1 )· ϕ ∈ Aut(R1 ) ϕ(B)
0
b and since R0 is the unique
The second equality holds since R1 = hR0 , Bi,
b =B
b , this,
abelian subgroup of R1 of index 2. Since γbu (R0 ) = R0 and γbu (B)
together with (3), shows that
b = B,
b ϕ|R ∈ AutF (R0 ) ,
AutF (R1 ) = Inn(R1 )· ϕ ∈ Aut(R1 ) ϕ(B)
0
and hence that
ϕ ∈ AutF (R1 ) ϕ(z) = z
b = B,
b ϕ(z) = z, ϕ|R ∈ AutF (R0 )
= Inn(R1 )· ϕ ∈ Aut(R1 ) ϕ(B)
0
b = B,
b ϕ|R ∈ AutSpin (R0 ) ,
= Inn(R1 )· ϕ ∈ Aut(R1 ) ϕ(B)
0
Geometry & Topology, Volume 9 (2005)
(6)
2410
where the second equality follows from (4). If ϕ ∈ Aut(R1 ) is such that
b = B
b and ϕ|R = cx |R for some x ∈ NSpin (R0 ), then x normalizes
ϕ(B)
0
0
b = cy (B)
b for some y ∈ R1 =
R1 (the centralizer of the 2–torsion in R0 ), cx (B)
b
R0 ·hBi by (5), so we can assume y ∈ R0 . Since R0 = A(q ∞ ) is abelian
(so [y, R0 ] = 1), this implies cy−1 x |R1 = ϕ, and hence ϕ ∈ AutSpin (R1 ). So
{ϕ ∈ AutF (R1 ) | ϕ(z) = z} ≤ AutSpin (R1 ) by (6), and the opposite inclusion is
clear.
Case 2 Now assume that E R0 . By assumption, U ≤ E (hence E ≤
CS (E) ≤ CS (U )), and CS (E) is a Sylow subgroup of CSpin (E). Also, E
contains an element of the form g·[[B i , B j , B k ]] for g ∈ R0 = A(q n ) and some
i, j, k not all even, and hence A(q n ) CS (E). Hence by (2), CS (E) is not
b
isomorphic to R1 = CS (hz, z1 , Ai),
and this shows that E is not Spin–conjugate
b
to hz, z1 , Ai. By [2, Proposition A.8], Spin contains exactly two conjugacy
classes of rank 3 subgroups containing z , and thus E must have type II. So by
[2, Proposition A.8(d)], CS (E) is elementary abelian of rank 4, and also has
type II.
b Bi
b ∼
Let C be the Spin7 (q n )–conjugacy class of the subgroup E∗ = hU, A,
= C24 ,
′
which by Lemma 1.8(a) has type I. Let E be the set of all subgroups of S which
are elementary abelian of rank 4, contain U , and are not in C . By Lemma
def
1.8(e), for any ϕ ∈ IsoΓ (E ′ , E ′′ ) and any E ′ ∈ E ′ , E ′′ = ϕ(E ′ ) ∈ E ′ , and ϕ
sends xC (E ′ ) to xC (E ′′ ). The same holds for ϕ ∈ IsoSpin (E ′ , E ′′ ) by definition
of the elements xC (−) ([2, Proposition A.9]). Since CS (E) ∈ E ′ , this shows
that all elements of AutF (CS (E)) send the element xC (CS (E)) to itself. By [2,
Proposition A.9(c)], AutSpin (CS (E)) is the group of automorphisms which are
the identity on the rank two subgroup hxC (CS (E)), zi; and (1) now follows.
The proof of the following lemma is essentially unchanged.
Lemma 1.10 [2, Lemma 2.10] Fix n ≥ 1, and let E, E ′ ≤ S(q n ) be two
elementary abelian subgroups of rank three which contain U , and which are
Γn –conjugate. Then E and E ′ are Spin7 (q n )–conjugate.
Proof Consider the sets
and
J1 = X ∈ SL2 (q n ) X 2 = −I
n
J2 = X ∈ SL2 (q 2n ) ψ q (X) = −X, X 2 = −I .
Geometry & Topology, Volume 9 (2005)
2411
n
n
Here, as usual, ψ q is induced by the field automorphism (x 7→ xq ). It was
shown in the proof of [2, Lemma 2.10] that all elements in each of these sets
are SL2 (q)–conjugate to each other.
Since E and E contain U , E, E ′ ≤ CSpin7 (qn ) (U ). By [2, Proposition 2.5(a)],
def
CSpin7 (qn ) (U ) = H(q n ) = ω(SL2 (q ∞ )3 ) ∩ Spin7 (q n ).
Thus
E = hz, z1 , [[X1 , X2 , X3 ]]i
and
E ′ = hz, z1 , [[X1′ , X2′ , X3′ ]]i,
where the Xi are all in J1 or all in J2 , and similarly for the Xi′ . Also, E and E ′
are Γn –conjugate, and each element of Γn leaves U = hz, z1 i and ω(SL2 (q n )3 )
invariant. Hence either E and E ′ are both contained in ω(SL2 (q n )3 ), in which
case the Xi and Xi′ are all in J1 ; or neither is contained in ω(SL2 (q n )3 ), in
which case the Xi and Xi′ are all in J2 . This shows that the Xi and Xi′ are
all SL2 (q n )–conjugate, and so E and E ′ are Spin7 (q n )–conjugate.
We are now ready to prove:
Proposition 1.11 [2, Proposition 2.11] For a fixed odd prime power q , let
S(q n ) ≤ S(q ∞ ) ≤ Spin7 (q ∞ ) be as defined above. Let z ∈ Z(Spin7 (q ∞ )) be
the central element of order 2. Then for each n, Fn = FSol (q n ) is saturated as
a fusion system over S(q n ), and satisfies the following conditions:
(a) For all P, Q ≤ S(q n ) which contain z , if α ∈ Hom(P, Q) is such that
α(z) = z , then α ∈ HomFn (P, Q) if and only if α ∈ HomSpin7 (qn ) (P, Q).
(b) CFn (z) = FS(qn ) (Spin7 (q n )) as fusion systems over S(q n ).
(c) All involutions of S(q n ) are Fn –conjugate.
Furthermore, Fm ⊆ Fn for m|n. The union of the Fn is thus a category
FSol (q ∞ ) whose objects are the finite subgroups of S(q ∞ ).
Proof We apply Proposition 1.3, where p = 2, G = Spin7 (q n ), S = S(q n ),
Z = hzi = Z(G), U = hz, z1 i, CG (U ) = H(q n ), S0 = CS (U ) = S0 (q n ), and
Γ = Γn ≤ Aut(S0 ). By Lemma 1.7, γbu preserves H(q n )–fusion in S0 . Since
Γ is generated by γbu and certain automorphisms of H(q n ), this shows that all
automorphisms in Γ preserve H(q n )–fusion.
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Condition (a) in Proposition 1.3 (all noncentral involutions in G are conjugate) holds since all subgroups in E2 are conjugate ([2, Proposition A.8]), and
condition (b) holds by definition of Γ. Condition (c) holds since
{γ ∈ Γ | γ(z) = z} = Inn(S0 (q n ))·hcτ i = AutNG (U ) (S0 (q n ))
by [2, Proposition 2.5(b)]. Condition (d) was shown in Proposition 1.9, and
condition (e) in Lemma 1.10. So by Proposition 1.3, Fn is a saturated fusion
system, and CFn (Z) = FS(qn ) (Spin7 (q n )).
The proofs of the other statements remain unchanged.
In Section 3 of [2], these corrections affect only the proof of Lemma 3.1. In
that proof, the groups E100 and E001 are not Γ1 –conjugate under the new
definitions; instead E100 is Γ1 –conjugate to a subgroup which is Spin7 (q)–
conjugate to E001 (and hence the two are F –conjugate). Also, when showing
that AutF (E001 ) is the group of all automorphisms which fix z = xC (E001 ),
it is important to know that all Γ1 –isomorphisms between subgroups in that
conjugacy class preserve the elements xC (−) (as shown in Lemma 1.8(e)), and
not just that Γ1 –automorphisms of E001 do so.
The changes do not affect the later sections. Just note that we are able to
consider FSol (q n ) as a subcategory of FSol (q kn ) for k > 1, because they were
b 2 )∗ in the definitions of
both chosen using the same “correction factor” u ∈ (Z
Γn and Γkn .
2
Proof of Proposition 1.3
Proposition 1.3 follows from Lemmas 1.3, 1.4, and 1.5 in [2], once they are
restated to assume the hypotheses of this new proposition, and not those of [2,
Proposition 1.2]. The only one of these lemmas whose proof is affected by the
change in hypotheses is Lemma 1.4, and so we restate and reprove it here.
Lemma 2.1 Assume the hypotheses of Proposition 1.3, and let
F = hFS (G); FS0 (Γ)i
be the fusion system generated by G and Γ. Then for all P, P ′ ≤ S which
contain Z ,
{ϕ ∈ HomF (P, P ′ ) | ϕ(Z) = Z} = HomG (P, P ′ ).
Geometry & Topology, Volume 9 (2005)
2413
Proof Upon replacing P ′ by ϕ(P ) ≤ P ′ , we can assume that ϕ is an isomorphism, and thus that it factors as a composite of isomorphisms
ϕ1
ϕ2
ϕ3
ϕk−1
ϕ
=
=
=
=
=
P = P0 −−∼
−→ P1 −−∼
−→ P2 −−∼
−→ · · · −−∼
−→ Pk−1 −−∼
−k→ Pk = P ′ ,
where for each i, ϕi ∈ HomG (Pi−1 , Pi ) or ϕi ∈ HomΓ (Pi−1 , Pi ). Let Zi ≤ Z(Pi )
be the subgroups of order p such that Z0 = Zk = Z and Zi = ϕi (Zi−1 ).
To simplify the discussion, we say that a morphism in F is of type (G) if it is
given by conjugation by an element of G, and of type (Γ) if it is the restriction
of an automorphism in Γ. More generally, we say that a morphism is of type
(G, Γ) if it is the composite of a morphism of type (G) followed by one of type
(Γ), etc. We regard IdP , for all P ≤ S , to be of both types, even if P S0 .
By definition, if any nonidentity isomorphism is of type (Γ), then its source
and image are both contained in S0 = CS (U ). In particular, if Pi S0 for
any 0 < i < k , then ϕi and ϕi+1 are both of type (G), and we can remove it
and replace the two morphisms by their composite. We can thus assume that
[Pi , U ] = 1 for all 0 < i < k , and hence that Zi ≤ Z(Pi U ) for all i.
For each i, using [2, Lemma 1.3], choose some ψi ∈ HomF (Pi U, S) such that
ψi (Zi ) = Z . More precisely, using points (1) and (2) in [2, Lemma 1.3], we
can choose ψi to be of type (Γ) if Zi ≤ U (the inclusion if Zi = Z ), and to
be of type (G, Γ) if Zi U . Set Pi′ = ψi (Pi ). To keep track of the effect of
morphisms on the subgroups Zi , we write them as morphisms between pairs,
as shown below. Thus, ϕ factors as a composite of isomorphisms
−1
ψi−1
ϕi
ψi
′
(Pi−1
, Z) −−−−−→ (Pi−1 , Zi−1 ) −−−−−→ (Pi , Zi ) −−−−−→ (Pi′ , Z).
If ϕi is of type (G), then this composite (after replacing adjacent morphisms
of the same type by their composite) is of type (Γ, G, Γ). If ϕi is of type (Γ),
then the composite is again of type (Γ, G, Γ) if either Zi−1 ≤ U or Zi ≤ U ,
and is of type (Γ, G, Γ, G, Γ) if neither Zi−1 nor Zi is contained in U . So we
are reduced to assuming that ϕ is of one of these two forms.
Case 1 Assume first that ϕ is of type (Γ, G, Γ); ie, a composite of isomorphisms of the form
ϕ1
ϕ2
ϕ3
(Γ)
(G)
(Γ)
(P0 , Z) −−−−→ (P1 , Z1 ) −−−−→ (P2 , Z2 ) −−−−→ (P3 , Z).
Then Z1 = Z if and only if Z2 = Z because ϕ2 is of type (G). If Z1 = Z2 = Z ,
then ϕ1 and ϕ3 are of type (G) by Proposition 1.3(c), and the result follows.
If Z1 6= Z 6= Z2 , then U = ZZ1 = ZZ2 , and thus ϕ2 (U ) = U . Neither ϕ1
nor ϕ3 can be the identity, so Pi ≤ S0 = CS (U ) for all i by definition of
Geometry & Topology, Volume 9 (2005)
2414
HomΓ (−, −). Let γ1 , γ3 ∈ Γ be such that ϕi = γi |Pi−1 , and let g ∈ NG (U ) be
such that ϕ2 = cg . Then gCS (U )g−1 ∈ Sylp (CG (U )), so there is h ∈ CG (U )
such that hg ∈ N (CS (U )). Then chg ∈ Γ by Proposition 1.3(c).
Since γ3 preserves CG (U )–fusion among subgroups of S0 , there is g′ ∈ CG (U )
−1
such that cg′ = γ3 ◦ c−1
h ◦ γ3 . Thus ϕ is the composite
ϕ1
chg
γ3
cg ′
(Γ)
(Γ)
(Γ)
(G)
P0 −−−−→ P1 −−−−→ hP2 h−1 −−−−→ γ3 (hP2 h−1 ) −−−−→ P3 .
The composite of the first three isomorphisms is of type (Γ) and sends Z to
g′−1 Zg′ = Z , hence is of type (G) by Proposition 1.3(c) again, and so ϕ is also
of type (G).
Case 2 Assume now that ϕ is of type (Γ, G, Γ, G, Γ); more precisely, that it
is a composite of the form
ϕ1
ϕ2
ϕ3
(Γ)
(G)
(Γ)
(P0 , Z) −−−→ (P1 , Z1 ) −−−→ (P2 , Z2 ) −−−→ (P3 , Z3 )
ϕ4
ϕ5
(G)
(Γ)
−−−→ (P4 , Z4 ) −−−→ (P5 , Z),
where Z2 , Z3 U . Then Z1 , Z4 ≤ U and are distinct from Z , and the
groups P0 , P1 , P4 , P5 all contain U since ϕ1 and ϕ5 (being of type (Γ)) leave
U invariant. In particular, P2 and P3 contain Z , since P1 and P4 do and
ϕ2 , ϕ4 are of type (G). We can also assume that U ≤ P2 , P3 , since otherwise
P2 ∩ U = Z or P3 ∩ U = Z , ϕ3 (Z) = Z , and hence ϕ3 is of type (G) by
Proposition 1.3(c) again. Finally, we assume that P2 , P3 ≤ S0 = CS (U ), since
otherwise ϕ3 = Id.
Let Ei ≤ Pi be the rank three elementary abelian subgroups defined by the
requirements that E2 = U Z2 , E3 = U Z3 , and ϕi (Ei−1 ) = Ei . In particular,
Ei ≤ Z(Pi ) for i = 2, 3 (since Zi ≤ Z(Pi ), and U ≤ Z(Pi ) by the above
remarks); and hence Ei ≤ Z(Pi ) for all i. Also, U = ZZ4 ≤ ϕ4 (E3 ) = E4 since
ϕ4 (Z) = Z , and thus U = ϕ5 (U ) ≤ E5 . Via similar considerations for E0 and
E1 , we see that U ≤ Ei for all i.
Set H = CG (U ) for short. Let E3 be the set of all elementary abelian subgroups
E ≤ S of rank three which contain U , and let E3∗ ⊆ E3 be the set of all E ∈ E3
such that CS (E) ∈ Sylp (CG (E)). Then for all E ∈ E3 ,
CS (E) = CS0 (E)
and
CG (E) = CH (E)
since E ≥ U (and S0 = CS (U )). Thus E3∗ is the set of all subgroups E ∈ E3
which are fully centralized in the fusion system FS0 (H), and so each subgroup
in E3 is H –conjugate to a subgroup in E3∗ .
Geometry & Topology, Volume 9 (2005)
2415
Let γ1 , γ3 , γ5 ∈ Γ be such that ϕi = γi |Pi−1 for odd i. Let g2 , g4 ∈ G be such
that ϕi is conjugation by gi for i = 2, 4. We will construct a commutative
diagram of the following form
cg
γ1
cg
γ3
γ5
2
4
(P2 , E2 ) → (P3 , E3 ) →
(P4 , E4 ) → (P5 , E5 )
(P0 , E0 ) → (P1 , E1 ) →
ca
ca0
ca
ca
ca
ca
2
4
1
3
5
↓
↓
↓
↓
↓
↓
c
c
γ
γ
γ
h
h
1
3
5
2
4
′
′
′
′
′
′
′
′
′
′
′
(C0 , E0 ) → (C1 , E1 ) → (C2 , E2 ) → (C3 , E3 ) → (C4 , E4 ) → (C5 , E5′ )
where Ei′ ∈ E3∗ , Ci′ = CS (Ei′ ), h2 , h4 ∈ G, and ai ∈ H = CG (U ). To do this,
def
first choose a′0 , a′2 , a′4 ∈ H such that Ei′ = a′i Ei a′i −1 ∈ E3∗ for i = 0, 2, 4, and set
′ ) for i = 1, 3, 5. Then C (E ′ ) = γ (C (E ′ )) for i = 1, 3, 5, so
Ei′ = γi (Ei−1
i
S
S
i−1
i
CS (Ei′ ) ∈ Sylp (CH (Ei′ )) and Ei′ ∈ E3∗ for all i. So we can choose x0 ∈ CH (E0′ )
′
′
such that x0 (a′0 P0 a′0 −1 )x−1
0 ≤ CS (E0 ), and set a0 = x0 a0 ∈ H . Since γ1 ∈
Aut(S0 ) preserves H –fusion, there is a1 ∈ H which makes the first square in
the above diagram commute. Now choose x2 ∈ CH (E2′ ) such that
−1 −1 −1
′
′
x2 (a′2 g2 a−1
x2 ≤ CS (E2′ ),
1 )CS (E1 )(a2 g2 a1 )
and set a2 = x2 a′2 and h2 = a2 g2 a−1
1 . Upon continuing this procedure we
obtain the above diagram.
Let ϕb ∈ IsoF (CS (E0′ ), CS (E5′ )) be the composite of the morphisms in the botb
tom row of the above diagram. Then ϕ(Z)
= Z , since ϕ and the cai all
send Z to itself. By Proposition 1.3(e), the rank three subgroups Ei′ are all
G–conjugate to each other. Choose g ∈ G such that gE5′ g−1 = E0′ . Then
g·CS (E5′ )·g−1 and CS (E0′ ) are both Sylow p–subgroups of CG (E0′ ), so there is
h ∈ CG (E0′ ) such that (hg)CS (E5′ )(hg)−1 = CS (E0′ ). By Proposition 1.3(d),
chg ◦ ϕb ∈ AutF (CS (E0′ )) is of type (G), so ϕb is of type (G), and hence
b ◦ ca0 is also of type (G).
ϕ = c−1
a5 ◦ ϕ
References
[1] W Dwyer, C Wilkerson, A new finite loop space at the prime two, J. Amer.
Math. Soc. 6 (1993) 37–64
[2] R Levi, B Oliver, Construction of 2–local finite groups of a type studied by
Solomon and Benson, Geom. Topol. 6 (2002) 917–990
[3] R Solomon, Finite groups with Sylow 2–subgroups of type 3, J. Algebra 28
(1974) 182–198
[4] M Suzuki, Group theory I, Grundlehren series 247, Springer–Verlag (1982)
Geometry & Topology, Volume 9 (2005)